Multiproduct static model with a limited capacity of the warehouse
From Supply Chain Management Encyclopedia
(→Inventory optimal control) |
|||
Line 1: | Line 1: | ||
+ | '''Russian: [http://scm.gsom.spbu.ru/index.php/Многопродуктовая_статическая_модель_с_ограниченной_вместимостью_склада Многопродуктовая статическая модель с ограниченной вместимостью склада]''' | ||
+ | |||
+ | |||
+ | Multiproduct static model with a limited capacity of the warehouse | ||
+ | |||
Consider the case of static demand. | Consider the case of static demand. | ||
Revision as of 11:59, 20 August 2011
Russian: Многопродуктовая статическая модель с ограниченной вместимостью склада
Multiproduct static model with a limited capacity of the warehouse
Consider the case of static demand.
The basic assumptions of the model
- consider the problem of managing multiple types of stock;
- storage space is limited;
- order cost is a constant;
- price of the resource unit is constant;
- carrying cost of resource unit is a constant;
- затраты на оформление, связанные с размещением заказа, - постоянная величина (константа);
- order cost is a constant;
- no deficit.
The basic notations
There are two types of inventory , :
- - demand rate;
- - marginal carrying costs;
- - fixed order costs;
- - total costs per unit of time;
- - order quantity;
- - economic order quantity;
- - the space required to store the unit;
- - the maximum space required to store types of resources.
Inventory optimal control
According to the assumptions of the model, consider dynamics of the resource stock , fig. 1:
Fig. 1.' Dynamics of changes in resource stock .
Multiproduct static model with a limited capacity of the warehouse can be formalized as a nonlinear programming problem:
.
Inventory optimal control
For the above problem of nonlinear programming Lagrange function has the form:
,
where is a Lagrange multiplier.
The Lagrange function for multiproduct static model with a limited capacity of the warehouse is convex, hence, the optimal value and can be found from the first order conditions:
(limitation on the capacity of a warehouse at the optimal point);
.
The solution of the second equation is:
.
The optimal solution value with the desired accuracy can be found as follows:
1. Set the initial value
2. Set the value for decreasing the value (accuracy)
3. Consistently reduce on the value of , substituting the value of in and checking the performance limitations on the capacity of the warehouse.
The optimal strategy for inventory management in our model has the form:
Step 1. Calculate the optimal volume of orders, not including the restriction on the capacity of storage (see Basic economic order quantity) as follows:
,
.
Step 2. Subject to the values found , verify constraints on the capacity of the warehouse. If this restriction is satisfied, then the set of values , is the optimal solution for multiproduct static model with a limited capacity of the warehouse. Otherwise, the best solution is the set ,